Exercises What are the solutions for each equation? Use the Quadratic Formula. \( \begin{array}{ll}\text { 1. }-x^{2}+7 x-3=0 & \text { 2. } x^{2}+6 x=10\end{array} \)

To solve each quadratic equation using the **Quadratic Formula**, we'll follow these steps: The **Quadratic Formula** is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). --- ### **1. Solving \(-x^2 + 7x - 3 = 0\)** First, identify the coefficients: \[ a = -1,\quad b = 7,\quad c = -3 \] **Step 1: Calculate the Discriminant (\( \Delta \))** \[ \Delta = b^2 - 4ac = 7^2 - 4(-1)(-3) = 49 - 12 = 37 \] **Step 2: Apply the Quadratic Formula** \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-7 \pm \sqrt{37}}{2(-1)} = \frac{-7 \pm \sqrt{37}}{-2} \] **Step 3: Simplify the Expression** Multiply numerator and denominator by \(-1\) to make the denominator positive: \[ x = \frac{7 \mp \sqrt{37}}{2} \] **Solutions:** \[ x = \frac{7 + \sqrt{37}}{2} \quad \text{and} \quad x = \frac{7 - \sqrt{37}}{2} \] --- ### **2. Solving \(x^2 + 6x = 10\)** First, rewrite the equation in standard form: \[ x^2 + 6x - 10 = 0 \] Identify the coefficients: \[ a = 1,\quad b = 6,\quad c = -10 \] **Step 1: Calculate the Discriminant (\( \Delta \))** \[ \Delta = b^2 - 4ac = 6^2 - 4(1)(-10) = 36 + 40 = 76 \] **Step 2: Apply the Quadratic Formula** \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-6 \pm \sqrt{76}}{2(1)} = \frac{-6 \pm \sqrt{76}}{2} \] **Step 3: Simplify the Square Root** \[ \sqrt{76} = \sqrt{4 \times 19} = 2\sqrt{19} \] **Step 4: Substitute Back into the Formula** \[ x = \frac{-6 \pm 2\sqrt{19}}{2} \] **Step 5: Simplify the Fraction** \[ x = \frac{-6}{2} \pm \frac{2\sqrt{19}}{2} = -3 \pm \sqrt{19} \] **Solutions:** \[ x = -3 + \sqrt{19} \quad \text{and} \quad x = -3 - \sqrt{19} \] --- ### **Summary of Solutions** 1. **For \(-x^2 + 7x - 3 = 0\):** \[ x = \frac{7 + \sqrt{37}}{2} \quad \text{and} \quad x = \frac{7 - \sqrt{37}}{2} \] 2. **For \(x^2 + 6x = 10\):** \[ x = -3 + \sqrt{19} \quad \text{and} \quad x = -3 - \sqrt{19} \] These are the exact solutions. If you need decimal approximations: - \(\sqrt{37} \approx 6.08276\), so: \[ x \approx \frac{7 + 6.08276}{2} \approx 6.54138 \quad \text{and} \quad x \approx \frac{7 - 6.08276}{2} \approx 0.45862 \] - \(\sqrt{19} \approx 4.3589\), so: \[ x \approx -3 + 4.3589 \approx 1.3589 \quad \text{and} \quad x \approx -3 - 4.3589 \approx -7.3589 \]